Asymptotic Enumeration of Dense 0-1 Matrices with Equal Row Sums and Equal Column Sums E. Rodney Canfield ∗ Department of Computer Science University of Georgia Athens, GA 30602, USA erc@cs.uga.edu Brendan D. McKay † Department of Computer Science Australian National University Canberra ACT 0200, Australia bdm@cs.anu.edu.au Submitted: Dec 22, 2004; Accepted: Jun 11, 2005; Published: Jun 19, 2005 Mathematics Subject Classifications: 05A16, 05C30, 62H17 Abstract Let s, t, m, n be positive integers such that sm = tn.LetB(m, s; n, t)bethe number of m × n matrices over {0, 1} with each row summing to s and each column summing to t. Equivalently, B(m, s; n, t) is the number of semiregular bipartite graphs with m vertices of degree s and n vertices of degree t. Define the density λ = s/n = t/m. The asymptotic value of B(m, s; n, t) has been much studied but the results are incomplete. McKay and Wang (2003) solved the sparse case λ(1−λ)=o  (mn) −1/2  using combinatorial methods. In this paper, we use analytic methods to solve the problem for two additional ranges. In one range the matrix is relatively square and the density is not too close to 0 or 1. In the other range, the matrix is far from square and the density is arbitrary. Interestingly, the asymptotic value of B(m, s; n, t) can be expressed by the same formula in all cases where it is known. Based on computation of the exact values for all m, n ≤ 30, we conjecture that the same formula holds whenever m + n →∞regardless of the density. 1 Introduction Let s, t, m, n be positive integers such that sm = tn.LetB(m, s; n, t)bethenumberof m × n matrices over {0, 1} with each row summing to s and each column summing to t. Equivalently, B(m, s; n, t) is the number of semiregular bipartite graphs with m vertices of degree s and n vertices of degree t.Thedensity λ = s/n = t/m is the fraction of entries in the matrix which are 1. ∗ Research supported by the NSA Mathematical Sciences Program † Research supported by the Australian Research Council the electronic journal of combinatorics 12 (2005), #R29 1 We are concerned in this paper with the asymptotic value of B(m, s; n, t). Historically, the first significant result was that of Read [20], who obtained the asymptotic behavior for s = t = 3. This was extended by Everett and Stein [8] to the case where s and t are arbitrary constants, not necessarily equal. The first result to allow s and t to increase was that of O’Neil [18], who permitted s, t = O  (log n) 1/4−  . This was improved by Mineev and Pavlov [17] to permit s = t ≤ γ(log n) 1/2 for fixed γ<1 and also for 1 <s≤ (t − 1) −1 γ(log n) 1/4 . McKay [13] obtained B(m, s; n, t) asymptotically whenever s, t = o  (sm) 1/4  .This was improved by McKay and Wang [14] to the case st = o  (mn) 1/2  . All the prior work so far mentioned considers matrices for which the density is quite small. Obviously B(m, n − s; n, m − t)=B(m, s; n, t) by complementation, so the very dense case is also handled. The intermediate range of densities, such as constant density, is considerably harder to deal with and until the present paper no exact asymptotics had been determined. Ordentlich and Roth [19] proved that, without any conditions except ms = nt, B(m, s; n, t) ≥  m t  n  n s  m  λ λ (1 − λ) 1−λ  mn , and that this bound is low by at most exp  O(n +logm)  uniformly over λ if λ(1 − λ)m exceeds some absolute constant. More recently, Litsyn and Shpunt [11] determined an upper bound on B(m, s; n, t)whenm =Θ(n)andλ = t/m = s/n is constant that, together with Ordentlich and Roth’s lower bound, gives that B(m, s; n, t)=  λ λ (1 − λ) 1−λ  −mn  2πλ(1 − λ)  −m/2−n/2 m −n/2 n −m/2 e O(n  ) for any >0. Without giving more than a heuristic justification, Good and Crook [9] suggested the approximation B(m, s; n, t) ≈  n s  m  m t  n  mn λmn  . We will see below that this is remarkably accurate, being within a constant of the correct value over a wide range and perhaps always. In the present paper, we will focus on two quite different cases, using analytic methods inspired by [15]. In one case, the matrix is relatively square and the density is not too close to 0 or 1. (This includes the range considered by Litsyn and Shpunt.) In the other case, the matrix is much wider than high (or vice-versa) but the density is arbitrary. In both cases, we obtain precise asymptotics. Remarkably, both the results we establish in this paper and the earlier results in the sparse case can be expressed using the same formula. Theorem 1. Consider a sequence of 4-tuples of positive integers m, s, n, t such that ms = nt and 1 ≤ t ≤ m − 1. Define λ = s/n = t/m and A = 1 2 λ(1 − λ). Suppose that >0 the electronic journal of combinatorics 12 (2005), #R29 2 is sufficiently small and that one of the following conditions holds (perhaps with m, n and s, t interchanged): (a) m, n →∞and st = o  (mn) 1/2  ; (b) m, n →∞with n ≤ m = o(A 2 n 1+ ) and, for some constant γ< 3 2 , (1 − 2λ) 2 m ≤ γAnlog n; (c) n →∞with 2 ≤ m = O  t(m − t)n  1/4−  . Then B(m, s; n, t)=  n s  m  m t  n  mn λmn   m − 1 m  (m−1)/2  n − 1 n  (n−1)/2 exp  1 2 + o(1)  . (1.1) Proof. Part (a) was established by McKay and Wang [14]. Part (b) will be proved in Sections 2–4; specifically, it follows from (2.2) and Theorems 2 and 3. Part (c) follows from Theorem 4 in Section 5. Note that  N − 1 N  (N−1)/2 =exp  − 1 2 + O(N −1 )  as N →∞, so one or both such terms in (1.1) can be simplified depending on which of m, n tend to ∞. In Section 6 we show how B(m, s; n, t) can be computed exactly for small m, n and show how the values for m, n ≤ 30 suggest the following conjecture. Conjecture 1. Consider a sequence of 4-tuples of positive integers m, s, n, t such that ms = nt. Then (1.1) holds uniformly over 1 ≤ t ≤ m − 1 whenever m + n →∞. Calculations of the exact values for all m, n ≤ 30 show excellent agreement with Con- jecture 1. There is less than 10% discrepancy between the exact value and the conjectured asymptotic value in all cases computed and less than 1% discrepancy whenever m+n ≥ 35. More precisely, write the quantity indicated by “o(1)” in (1.1) as ∆(m, s; n, t)/(Amn). Our experiments, including the exact values mentioned above and many numerical esti- mates described in Section 6, suggest that ∆(m, s; n, t) always lies in the interval (− 1 12 , 0). From [14], (see [10, Corollary 5.1]), we know that ∆(m, s; n, t) →− 1 12 as m, n →∞with st = o  (mn) 1/5  . At the upper end, the greatest value we know is ∆(4, 2; 4, 2) ≈−0.0171. In a future paper we will allow the row sums, and similarly the column sums, to be unequal within limits. For the case of sparse matrices, the best result is by Greenhill, McKay and Wang [10]. We also plan to address the issue of matrices over {0, 1, 2, } with equal row sums and equal column sums. the electronic journal of combinatorics 12 (2005), #R29 3 2AnintegralforB(m, s; n, t) Our proof of Theorem 1(b) occupies this section and the following two. We express B(m, s; n, t)asanintegralin(m+n)-dimensional complex space then estimate its value by the saddle-point method. It is clear that B = B(m, s; n, t) is the coefficient of x s 1 ···x s m y t 1 ···y t n in m  j=1 n  k=1  1+x j y k  . Applying Cauchy’s Theorem we have B = 1 (2πi) m+n  ···   j,k (1 + x j y k ) x s+1 1 ···x s+1 m y t+1 1 ···y t+1 n dx 1 ···dx m dy 1 ···dy n , (2.1) where each contour circles the origin once in the anticlockwise direction. It will suffice to take the contours to be circles; specifically, we will put x j = re iθ j and y k = re iφ k for each j, k,where r =  λ 1 − λ . This gives B = 1 (2π) m+n  λ λ (1 − λ) 1−λ  mn I(m, n), (2.2) where I(m, n)=  π −π ···  π −π  j,k  1+λ(e i(θ j +φ k ) − 1)  e is j θ j +it k φ k dθ dφ, (2.3) where θ =(θ 1 , ,θ m )andφ =(φ 1 , ,φ n ). In equation (2.3) it is to be noted that the integrand is invariant under the two substitutions θ j ← θ j +2π and φ k ← φ k +2π. In analyzing the magnitude of this integrand, it is often necessary to consider what might be called the “wrap-around” neighborhood of apointθ ∈ [−π, +π]. This neighborhood consists of the union of two half-open intervals [−π, −π + δ)and(π − δ, π]. To avoid numerous awkward expressions such as this, we find it convenient to think of θ j and φ k as points on the unit circle. To this end, we let C be the real numbers modulo 2π, which we can interpret as points on a circle in the usual fashion. Let z be the canonical mapping from C to the real interval (−π,π]; that is, if x lies on the unit circle, then z(x) is its signed arc length from the point 1. An open half-circle is C t =(t − π/2,t+ π/2) ⊆ C for some t. With this notion of half-circle, we may define an important subset of the Cartesian product C N ; namely, define ˆ C N to be the subset of vectors x =(x 1 , ,x N ) ∈ C N such that x 1 , ,x N all lie in a single open half-circle (where that open half-circle can depend on x). the electronic journal of combinatorics 12 (2005), #R29 4 If x =(x 1 , ,x N ) ∈ C N 0 then define ¯ x = z −1  1 N N  j=1 z(x j )  . More generally, if x ∈ C N t then define ¯ x = t + (x 1 − t, ,x N − t). It is easy to see that the function x → ¯ x is well-defined and continuous for x ∈ ˆ C N . 3 The principal part of the integral To estimate the integral I(m, n), we show that it is concentrated in a rather small region, then we expand the integrand inside that region. For some sufficiently small >0, let R denote the set of vector pairs θ, φ ∈ ˆ C m × ˆ C n such that | ¯ θ + ¯ φ|≤(mn) −1/2+2 | ˆ θ j |≤n −1/2+ , 1 ≤ j ≤ m | ˆ φ k |≤m −1/2+ , 1 ≤ k ≤ n, where ˆ θ j = θ j − ¯ θ and ˆ φ k = φ k − ¯ φ. In this definition, values are considered in C. Let I R (m, n) denote the integral I(m, n) restricted to the region R. In the following section, we will show that I(m, n) ∼ I R (m, n). In the present section, we will estimate I R (m, n). Our calculations are guided by the similar problem solved in [15]. In particular, we will use the following result which can be proved from a special case of [15, Lemma 3]. Let Im(z) denote the imaginary part of z. Lemma 1. Let  and   be such that 0 <  < 2< 1 12 .Let ˆ A = ˆ A(N) be a real- valued function such that N −  ≤ ˆ A(N) ≤ N   for sufficiently large N.Let ˆ B = ˆ B(N), ˆ C = ˆ C(N), ˆ E = ˆ E(N), ˆ F = ˆ F (N) be complex-valued functions such that the ratios ˆ B/ ˆ A, ˆ C/ ˆ A, ˆ E/ ˆ A, ˆ F/ ˆ A are bounded. Suppose that, for some δ>0, f(z)=exp  − ˆ ANξ 2 + ˆ BNξ 3 + ˆ Cξ 1 ξ 2 + ˆ ENξ 4 + ˆ Fξ 2 2 + O(N −δ )  is integrable for z =(z 1 ,z 2 , ,z N ) ∈ U N , where ξ t =  N j=1 z t j for t =1, 2, 3, 4 and U N =  z   |z j |≤N −1/2+ for 1 ≤ j ≤ N  . Then, provided the O() term in the following converges to zero,  U N f(z) dz =  π ˆ AN  N/2 exp  3 ˆ E + ˆ F 4 ˆ A 2 + 15 ˆ B 2 +6 ˆ B ˆ C + ˆ C 2 16 ˆ A 3 + O  (N −1/2+12 + N −δ ) ˆ Z + ˆ A −1 N − 1 4 +3   , the electronic journal of combinatorics 12 (2005), #R29 5 where ˆ Z =exp  15 Im( ˆ B) 2 +6Im( ˆ B)Im( ˆ C)+Im( ˆ C) 2 16 ˆ A 3  . Proof. Lemma 3 of [15] implies a result that is the same except that the condition N −  ≤ ˆ A(N) ≤ N   is replaced by the stronger condition N −  ≤ ˆ A(N)=O(1) and the condition < 1 24 is is replaced by the weaker condition < 1 12 . Moreover, the error term is O  (N −1/2+6 + N −δ ) ˆ Z + N −1+12 + ˆ A −1 N −∆  for any ∆ satisfying 0 < ∆ < 1 4 − 1 2 . Clearly this covers the case N −  ≤ ˆ A(N) ≤ 1of the present lemma, on taking ∆ = 1 4 − . For the remaining case, where 1 ≤ ˆ A(N) <N   , apply the transformation z j → N −  /2 z j , then invoke Lemma 3 of [15] again, using ∆ = 1 4 −  as before. In the following, we assume that m, n →∞. A word of explanation about the symbol  as used in the paper is in order. It represents a definite positive constant. Whenever an assertion is made which the reader can confirm only by knowing the value of ,s/he should note that the assertion is correct as long as  is small enough. There being only finitely many statements in the paper, there is some positive value for  small enough for all of them. In short, all equations and inequalities should be read with an understood “for m, n sufficiently large and  sufficiently small”. The following lemma will be needed soon. We use the notation R c for the complement of aregionR. Recall that A = 1 2 λ(1 − λ). Lemma 2. Let m, n →∞be integers, x 1 , ,x m variables, M 2 =  m j=1 x 2 j , and K the region of m-space defined by K =  x    m 2An (1 − m −1/4 ) ≤ M 2 ≤ m 2An (1 + m −1/4 )  . Then,  K c exp(−AnM 2 ) dx = O(1)  π An  m/2 exp  − 1 5 m 1/2  . Proof. We’ll be brief, because the idea is very much the same as found in the proof of Lemma 1, which can be consulted for details in [15]. Recalling the formula for the surface area of the ball of radius ρ in m-space, we have  M 2 ∈[a,b] exp(−AnM 2 )= 2π m/2 Γ(m/2)  b 1/2 a 1/2 e −Anρ 2 ρ m−1 dρ. Case (i): a =0,b =(m/(2An))(1 − m −1/4 ). Using e −An(b−x) 2 (b − x) m−1 ≤ e −Anb 2 −Anx 2 b m−1 , 0 ≤ x ≤ b, the electronic journal of combinatorics 12 (2005), #R29 6 and Stirling’s formula for the Gamma function,  M 2 ∈[0,b] exp(−AnM 2 )=O(1)  π An  m/2 exp  − 1 5 m 1/2  . Case (ii): a =(m/(2An))(1 + m −1/4 ), b = ∞.Using e −An(a+x) 2 (a + x) m−1 ≤ e −Ana 2 −Anx 2 a m−1 ,x≥ 0, we find the same bound for the integral over M 2 ∈ [a, ∞) as in Case (i). Combining the two cases completes the proof of the Lemma. Let T 1 be the transformation which expresses the original m + n variables θ j ,φ k (see (2.3)) in terms of µ = ¯ θ + ¯ φ, δ = ¯ θ − ¯ φ, ˆ θ j (1 ≤ j ≤ m − 1), and ˆ φ k (1 ≤ k ≤ n − 1). Explicitly, θ j = 1 2 (µ + δ)+ ˆ θ j ,φ k = 1 2 (µ − δ)+ ˆ φ k , where here and hereafter we use the abbreviations ˆ θ m = − m−1  j=1 ˆ θ j , ˆ φ n = − n−1  k=1 ˆ φ k . We have I R (m, n)=2πmnJ(m, n), where J(m, n)=  S G(µ, ˆ θ, ˆ φ) d ˆ θ d ˆ φ dµ. Here, the function G is the composition F ◦ T 1 , which is easily seen to be independent of the difference δ = ¯ θ − ¯ φ. The region of integration S = T −1 1 (R) is defined by virtually the same inequalities as was R with these two notes: we now write the first inequality as |µ|≤(mn) −1/2+2 ; and, second, neither ˆ θ m nor ˆ φ n is a variable of integration, but the definition of S includes the inequalities    m−1  j=1 ˆ θ j    ≤ n −1/2+ ,    n−1  k=1 ˆ φ k    ≤ m −1/2+ arising from the R-inequalities | ˆ θ m |≤n −1/2+ and | ˆ φ n |≤m −1/2+ . The factor of 2πmn comes from the integration over δ (which has a range of 4π) and the Jacobian mn/2of transformation T 1 . In this section we prove the electronic journal of combinatorics 12 (2005), #R29 7 Theorem 2. Suppose m, n →∞with λ = λ(m, n), such that m ≥ n and m = o(A 2 n 1+ ). (3.1) Suppose further that, for some constant γ< 3 2 − 45 2  − 6 2 , (1 − 2λ) 2 m ≤ γAnlog n. (3.2) Then, J(m, n)=(mn) −1/2 exp  − 1 2 − 1 − 2A 24A  m n + n m  + O(D)  ×  π Amn  1/2  π An  (m−1)/2  π Am  (n−1)/2 , where D = n −1/4+γ/24+4+o(1) + n −1/2+γ/3+15/2+2 2 . Proof. The assumption m ≥ n has been made only to avoid frequent use of the expressions max(m, n) and min(m, n). Two easy consequences of (3.1) will be used without repeatedly citing that equation: A −1 ≤ A −1 m n = o(An  ),m= o(An 1+ ). For future reference we establish: log n = o(An  ), log m = o(Am  ). (3.3) Indeed, for the first, log 2 n = o(A −1 · An  ), and A −1 = O(An  ). The second then follows since log m = O(log n)andm ≥ n. In particular, both Am  ,An  become infinite. For |x| small, see [15], 1+λ(e ix − 1) = exp  λix − Ax 2 − iA 3 x 3 + A 4 x 4 + O(A|x| 5 )  with A = 1 2 λ(1 − λ),A 3 = 1 6 λ(1 − λ)(1 − 2λ),A 4 = 1 24 λ(1 − λ)(1 − 6λ +6λ 2 ). Uniformly in the region S,whereall|µ + ˆ θ j + ˆ φ k | are small, G =exp  −A  j,k (µ + ˆ θ j + ˆ φ k ) 2 − iA 3  j,k (µ + ˆ θ j + ˆ φ k ) 3 + A 4  j,k (µ + ˆ θ j + ˆ φ k ) 4 + O  A  j,k |µ + ˆ θ j + ˆ φ k | 5   . Here and below, the undelimited summation over j, k runs over 1 ≤ j ≤ m,1≤ k ≤ n, and we continue to use the abbreviations ˆ θ m = −  m−1 j=1 ˆ θ j , ˆ φ n = −  n−1 k=1 ˆ φ k . the electronic journal of combinatorics 12 (2005), #R29 8 We now proceed to a second change of variables, ( ˆ θ, ˆ φ)=T 2 (σ, τ )givenby ˆ θ j = σ j + cµ 1 , ˆ φ k = τ k + dν 1 , where, for 1 ≤ h ≤ 4, µ h and ν h denote the power sums  m−1 j=1 σ h j and  n−1 k=1 τ h k , respec- tively. The scalars c and d are chosen to eliminate the second-degree cross-terms σ j 1 σ j 2 and τ k 1 τ k 2 , and thus diagonalize the quadratic in σ, τ . Suitable choices for c, d are c = − 1 m + m 1/2 ,d= − 1 n + n 1/2 , and we find the following:  j,k (µ + ˆ θ j + ˆ φ k ) 2 = mnµ 2 + nµ 2 + mν 2  j,k (µ + ˆ θ j + ˆ φ k ) 3 = mnµ 3 +3µ(nµ 2 + mν 2 )+n(µ 3 +3cµ 2 µ 1 − c 2 µ 3 1 ) + m(ν 3 +3dν 2 ν 1 − d 2 ν 3 1 )  j,k (µ + ˆ θ j + ˆ φ k ) 4 = mnµ 4 +6µ 2 ν 2 + n(µ 4 +4cµ 3 µ 1 +6c 2 µ 2 µ 2 1 + c 3 µ 4 1 ) + m(ν 4 +4dν 3 ν 1 +6d 2 ν 2 ν 2 1 + d 3 ν 4 1 )+6µ 2 (nµ 2 + mν 2 ) +4µ  n(µ 3 +3cµ 2 µ 1 − c 2 µ 3 1 )+m(ν 3 +3dν 2 ν 1 − d 2 ν 3 1 )   j,k |µ + ˆ θ j + ˆ φ k | 5 = O(mn −3/2+5 + nm −3/2+5 ), in which we have introduced the additional abbreviations c 2 = 1 m 1/2 (m 1/2 +1) 2 ,c 3 = m 1/2 +3 m(m 1/2 +1) 3 , d 2 = 1 n 1/2 (n 1/2 +1) 2 ,d 3 = n 1/2 +3 n(n 1/2 +1) 3 . The determinant of the matrix T 2 is (mn) −1/2 ,andso J(m, n)=(mn) −1/2  T −1 2 (S) E 1 , where E 1 =exp(L 1 ), and L 1 = µ 4 (A 4 mn)+µ 3 (−iA 3 mn)+µ 2 (−Amn +6A 4 nµ 2 +6A 4 mν 2 ) + µ  −3iA 3 nµ 2 − 3iA 3 mν 2 +4A 4 n(µ 3 +3cµ 2 µ 1 − c 2 µ 3 1 ) +4A 4 m(ν 3 +3dν 2 ν 1 − d 2 ν 3 1 )  − Anµ 2 − Amν 2 +6A 4 µ 2 ν 2 − iA 3 n(µ 3 +3cµ 2 µ 1 − c 2 µ 3 1 ) − iA 3 m(ν 3 +3dν 2 ν 1 − d 2 ν 3 1 ) + A 4 n(µ 4 +4cµ 3 µ 1 +6c 2 µ 2 µ 2 1 + c 3 µ 4 1 )+A 4 m(ν 4 +4dν 3 ν 1 +6d 2 ν 2 ν 2 1 + d 3 ν 4 1 ) + O(Amn −3/2+5 + Am −3/2+5 n). the electronic journal of combinatorics 12 (2005), #R29 9 To complete the evaluation of the integral, we need to consider a number of differ- ent regions within the space of the variables µ, σ j ,τ k ,aswellasanumberofdifferent integrands. Let us introduce all of these at the outset. Define ρ σ ,ρ τ > 0by ρ 2 σ = m 2An ,ρ 2 τ = n 2Am . The regions we shall use, in addition to T −1 2 (S), are these: Q =  |σ j |≤n −1/2+ ,j=1, ,m−1  ∩  |τ k |≤m −1/2+ ,k=1, ,n−1  ∩  |µ|≤(mn) −1/2+2  M =  |µ 1 |≤m 1/2 n −1/2+  ∩  |ν 1 |≤n 1/2 m −1/2+  B =  (1 − m −1/4 )ρ 2 σ ≤ µ 2 ≤ (1 + m −1/4 )ρ 2 σ  ∩  (1 − n −1/4 )ρ 2 τ ≤ ν 2 ≤ (1 + n −1/4 )ρ 2 τ  . As integrands we will use three functions E h =exp(L h ), h =1, 2, 3. The definition of L 1 has appeared already. The function L 2 consists of some of the summands found in L 1 : L 2 = −Amnµ 2 +6A 4 µ 2 ν 2 + A 4 nµ 4 + A 4 mν 4 − 3iA 3 nµµ 2 − 3iA 3 mµν 2 − Anµ 2 − Amν 2 − iA 3 nµ 3 − iA 3 mν 3 − 3iA 3 cnµ 2 µ 1 − 3iA 3 dmν 2 ν 1 . The third function L 3 equals Re(L 2 ), the real part of L 2 : L 3 = −Amnµ 2 +6A 4 µ 2 ν 2 + A 4 nµ 4 + A 4 mν 4 − Anµ 2 − Amν 2 . For convenience we define two expressions in m, n that recur in our big-oh expressions, H 1 = Am 1/2+2 n −1+5 + An 1/2+2 m −1+5 H 2 = A(mn) 2 + Amn −1+4 + Am −1+4 n. Having made all the necessary definitions, the next step is to establish a few relationships among the regions and functions just defined. Summing for 1 ≤ j ≤ m − 1 the equation ˆ θ j = σ j + cµ 1 , and inserting the value of c, we find m −1/2 µ 1 = m−1  j=1 ˆ θ j . In the region S we have |  m−1 j=1 ˆ θ j |≤n −1/2+ ,andsoinT −1 2 (S)wehave |µ 1 |≤m 1/2 n −1/2+ . Similarly, |ν 1 |≤n 1/2 m −1/2+ ; using these, the reader can check that 1 2 Q∩M⊆T −1 2 (S) ⊆ 3 2 Q∩M. the electronic journal of combinatorics 12 (2005), #R29 10 [...]... 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Asymptotic enumeration of regular matrices, e e e e o Studia Sci Math Hungar., 7 (1972) 343–353 [2] E A Bender, The asymptotic number of nonnegative integer matrices with given row and column sums, Discrete Math., 10 (1974) 345–353 [3] R N Bhattacharya and R R Rao, Normal Approximation and Asymptotic Expansions, John Wiley & Sons (NY, 1976) [4] B Bollob´s and B D McKay, The number of matchings in random... are equal and positive while half are equal and negative, and minimized when one entry is positive and the rest are equal and negative (or vice-versa) This gives 1 m2 − 3m + 3 π4 (x) ≤ ≤ m π2 (x)2 m(m − 1) The required inequality now follows ˆ Define U to be the set of vectors θ ∈ C m such that ˆ |θj | ≤ (An)−1/2+ /4 , 1 ≤ j ≤ m, ˆ ¯ where θj = θj − θ and A = 1 λ(1 − λ) as before 2 Lemma 6 If m ≥ 4 and. .. For small values of m and n, we can extract the constant term of G by using a method of summing over roots of unity A technique of this nature was given by Good and Crook [9] and improved by McKay [12] We will further improve it in this paper Let q1 and q2 be integers such that q1 ≥ m − t + 1 and q2 ≥ s + 1 Consider any field F which contains elements α, β of multiplicative order q1 and q2 , respectively... journal of combinatorics 12 (2005), #R29 29 If we define u(1) = p(1) , then the matrix with columns u(1) , u(2) , , u(n) has all column sums t and row sums s It is not a uniform sample from this class of matrices, but nevertheless the method can be extended to estimate any statistic on the class See [6] for details To further test Conjecture 1, we used this method to obtain accurate estimates of B(m,... 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Statist Assoc., 100 (2005) 109–120 [7] S X Chen and J S Liu, Statistical Applications of the Poisson-Binomial and conditional Bernoulli distributions, Statistica Sinica, 7 (1997) 875–892 [8] C J Everett, Jr., and P R Stein, The asymptotic number of integer stochastic matrices, Discrete Math., 1 (1971) 33–72 [9] I J Good and J F Crook, The enumeration of arrays and a generalization related to contingency... decreased if xj and xk are moved slightly towards each other, which can be done within the constraints on x unless |x | = 1 for exactly one value of , and the other entries of x are equal This therefore locates the minimum of π2 (x) The location of the maximum can be similarly identified, but it is easier to just note that π2 (x) ≤ m trivially For the second line of (b), we work similarly with π3 (x) If . Asymptotic Enumeration of Dense 0-1 Matrices with Equal Row Sums and Equal Column Sums E. Rodney Canfield ∗ Department of Computer Science University of Georgia Athens, GA 30602,. the case of sparse matrices, the best result is by Greenhill, McKay and Wang [10]. We also plan to address the issue of matrices over {0, 1, 2, } with equal row sums and equal column sums. the. t)bethenumberof m × n matrices over {0, 1} with each row summing to s and each column summing to t. Equivalently, B(m, s; n, t) is the number of semiregular bipartite graphs with m vertices of degree s and